Tricritical Dicke model with and without dissipation

Light-matter interacting systems involving multilevel atoms are appealing platforms for testing equilibrium and dynamical phenomena. Here we explore a tricritical Dicke model, where an ensemble of three-level systems interacts with a single light mode, through two different approaches: a generalized Holstein-Primakoff mapping and a treatment using the Gell-Mann matrices. Both methods are found to be equivalent in the thermodynamic limit. In equilibrium, the system exhibits a rich phase diagram where both continuous and discrete symmetries can be spontaneously broken. We characterize all the different types of symmetries according to their scaling behaviors. Far from the thermodynamic limit, considering just a few tens of atoms, the system already exhibits features that could help characterize both second- and first-order transitions in a potential experiment. Importantly, we show that the tricritical behavior is preserved when dissipation is taken into account. Moreover, the system develops a rich steady-state phase diagram with various regions of bistability, all of them converging at the tricritical point. Having multiple stable normal and superradiant phases opens prospective avenues for engineering interesting steady states by a proper choice of initial states and/or parameter quenching.